A Strategy for Removing the Bias in the Graphical Analysis Method

The graphical analysis method, which transforms multiple time measurements of plasma and tissue uptake data into a linear plot, is a useful tool for rapidly obtaining information about the binding of radioligands used in PET studies. The strength of the method is that it does not require a particular model structure. However, a bias is introduced in the case of noisy data resulting in the underestimation of the distribution volume (DV), the slope obtained from the graphical method. To remove the bias, a modification of the method developed by Feng et al. (1993), the generalized linear least squares (GLLS) method, which provides unbiased estimates for compartment models was used. The one compartment GLLS method has a relatively simple form, which was used to estimate the DV directly and as a smoothing technique for more general classes of model structures. In the latter case, the GLLS method was applied to the data in two parts, that is, one set of parameters was determined for times 0 to T1 and a second set from T1 to the end time. The curve generated from these two sets of parameters then was used as input to the graphical method. This has been tested using simulations of data similar to that of the PET ligand [11C]-d-threo-methylphenidate (MP, DV = 35 mL/mL) and 11C raclopride (RAC, DV = 1.92 mL/mL) and compared with two examples from image data with the same tracers. The noise model was based on counting statistics through the half-life of the isotope and the scanning time. Five hundred data sets at each noise level were analyzed. Results (DV) for the graphical analysis (DV g ), the nonlinear least squares (NLS) method (DV nls ), the one-tissue compartment GLLS method (DV f ), and the two part GLLS followed by graphical analysis (DV fg ) were compared. DVFG was found to increase somewhat with increasing noise and in some data sets at high noise levels no estimate could be obtained. However, at intermediate levels it provided a good estimation of the true DV. This method was extended to use a reference tissue in place of the input function to generate the distribution volume ratio (DVR) to the reference region. A linearized form of the simplified reference tissue method of Lammertsma and Hume (1996) was used. The DVR generated directly from the model (DVR fl ) was compared with DVR fg (determined from a “smoothed” uptake curve as for DV fg ) using the graphical method.